The theory of determinants in the historical order of development, by Sir Thomas Muir.

12 ITISTORY:: OF THE THEORY OF DETERMINANTS

and (2), (3) and (4), (5) and (6), lie in the same plane, may be written
down under the form.*
A A2 A3 A4. =0.
B1 B2 B5 B4
Cl C2 C3 C4
A3 A4 A A5
B. B4 B5 B6
C3 C4 C, C,
"Consider now the points 1, 2, 3, 4, 5, 6, the coordinates of these
being respectively x1,.Y, xl......, x6, Y6, Z6. I represent, for shortness, the equation to the plane passing through the origin, and the
points 1, 2, which may be called the- plane 12, in the form
x 12x +y 12Y + z 12" =o;
consequently the -symbols 12,, 12, 12, denote respectively y~z2 -Zx3 - x xlyS xy1, and similarly for the. planes 13, &c. If now the
intersections of 12 and 45, 23 and 56, 34 and 61 lie in the same plane,
we must have by lemma (1) the equation
12X 45X 23X 56X.. =0.
12, 45- -23, 56,;12;, 45Z 23Z 56Z
23x 56X 34X 61X
23, 56, 34, 61
23Z 56Z 34z 61z
Multiplying the two sides of this equation by the two sides re-,
spectively of the equation
6.x i*= 612.345,
ys Yi Y2
Z6 zi 1:
* 4 X5. ys y4 Y5
and observing the equations
xT612x + Y612 5 + z612z 612, 1 2 0, &c.
* The commas which Cayley prints after tile elements in a'determinant we
omit here and henceforth.